User:Tmhoang81/Dynamic stability of discrete systems

Let us consider the discrete system, reference

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Kinetic energy:

$\displaystyle T={\frac {1}{2}}m{\dot {x}}_{1}^{2}+{\frac {1}{2}}m{\dot {x}}_{2}^{2}$

(1)

Potential energy:

$\displaystyle V={\frac {1}{2}}kx_{1}^{2}+{\frac {1}{2}}k_{2}x_{2}^{2}+{\frac {1}{2}}k_{n}(x_{2}-x_{1})^{2}$

(2)

The dissipation energy:

$\displaystyle W=-c_{1}{\dot {x}}_{1}x_{1}-c_{2}({\dot {x}}_{2}-{\dot {x}}_{1})(x_{2}-x_{1})-c_{3}{\dot {x}}_{2}x_{2}$

(3)

and then write the equations of motion using Lagrange's equations:

$\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {x}}_{i}}}-{\frac {\partial T}{\partial x_{i}}}+{\frac {\partial V}{\partial x_{i}}}=Q_{i},\qquad i=1,2$

(4)

where generalzied forces:

$\displaystyle Q_{i}={\frac {\partial W}{\partial x_{i}}}$

(5)

Substitute (1),(2),(3) and (5) to (4) to get:

$\displaystyle m{\ddot {x}}_{1}+\left[kx_{1}-k_{n}(x_{2}-x_{1})\right]=-c_{1}{\dot {x}}_{1}+c_{2}({\dot {x}}_{2}-{\dot {x}}_{1})$

(6)

$\displaystyle m{\ddot {x}}_{2}+\left[k_{2}x_{2}+k_{n}(x_{2}-x_{1})\right]=-c_{2}({\dot {x}}_{2}-{\dot {x}}_{1})-c_{3}{\dot {x}}_{2}$

(7)

or

$\displaystyle m{\ddot {x}}_{1}+(c_{1}+c_{2}){\dot {x}}_{1}-c_{2}{\dot {x}}_{2}+(k_{n}+k)x_{1}-k_{n}x_{2}=0$

(8)

$\displaystyle m{\ddot {x}}_{2}-c_{2}{\dot {x}}_{1}+(c_{2}+c_{3}){\dot {x}}_{2}-k_{n}x_{1}+(k_{n}+k_{2})x_{2}=0$

(9)

Assume the solutions of (8) and (9) having the following normal modes:

$\displaystyle x_{1}=A_{1}e^{\omega t},\qquad x_{2}=A_{2}e^{\omega t}$

(10)

Plugging (10) to (8) and (9):

$\displaystyle [k_{n}+k+(c_{1}+c_{2})\omega +m\omega ^{2}]A_{1}-(k_{n}+\omega c_{2})A_{2}=0$

(11)

$\displaystyle -(k_{n}+\omega c_{2})A_{1}+[k_{n}+k_{2}+(c_{2}+c_{3})\omega +m\omega ^{2}]A_{2}=0$

(12)

(11) and (12) have non-trial solutions if the determinant of the coefficients of $\displaystyle A_{i}(i=1,2)$ is zero:

$\displaystyle [k_{n}+k+(c_{1}+c_{2})\omega +m\omega ^{2}][k_{n}+k_{2}+(c_{2}+c_{3})\omega +m\omega ^{2}]-(k_{n}+\omega c_{2})^{2}=0$

(13)

or rewrite this in the form of unknown $\displaystyle \omega$

$\displaystyle m^{2}\omega ^{4}+(c_{1}+2c_{2}+c_{3})m\omega ^{3}+[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}+(k+k_{2}+2k_{n})m]\omega ^{2}+[c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})]\omega +kk_{2}+(k+k_{2})k_{n}=0$

(14)

(8) and (9) have bounded solutions as time goes to infinity if $\displaystyle Re(\omega )<0$. We will use this condition to impose conditions on the coefficients of equation (14). Before doing this, let us review a famous Hurwitz's theorem, reference,

An equation of order n

$\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0$

(15)

has all solutions (complex or real) with negative parts if the following conditions are satisfied:

$\displaystyle \Delta _{1}=a_{1}>0,\quad \Delta _{2}={\begin{vmatrix}a_{1}&a_{0}\\a_{3}&a_{2}\end{vmatrix}}>0,\quad \Delta _{3}={\begin{vmatrix}a_{1}&a_{0}&0\\a_{3}&a_{2}&a_{1}\\a_{5}&a_{4}&a_{3}\end{vmatrix}}>0,\quad \cdots ,\quad \Delta _{n}={\begin{vmatrix}a_{1}&a_{0}&0&0&\cdots &0\\a_{3}&a_{2}&a_{1}&a_{0}&\cdots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{2n-1}&a_{2n-2}&a_{2n-3}&a_{2n-4}&\cdots &a_{n}\end{vmatrix}}=a_{n}\Delta _{n-1}>0$

(16)

Apply this condition for the 4th order equation:

$\displaystyle a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}x+a_{4}=0$

(17)

we obtain:

$\displaystyle a_{1}>0,\quad (a_{1}a_{2}-a_{0}a_{3})>0,\quad a_{3}(a_{1}a_{2}-a_{0}a_{3})-a_{4}a_{1}^{2}>0,\quad a_{4}>0$

(18)

It is easy to prove that (18) is equivalent to the following condition:

$\displaystyle a_{1},a_{3},a_{4}>0,\quad \quad a_{3}(a_{1}a_{2}-a_{0}a_{3})-a_{4}a_{1}^{2}>0$

(19)

Now require (19) must be true for coefficients of equation (14)

$\displaystyle {\begin{aligned}&m(c_{1}+2c_{2}+c_{3})>0\\&c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})>0\\&kk_{2}+(k+k_{2})k_{n}>0\end{aligned}}$

(20)

and

$\displaystyle {\begin{Bmatrix}[c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})]{\begin{pmatrix}m(c_{1}+2c_{2}+c_{3})[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}+(k+k_{2}+2k_{n})m]\\-m^{2}[c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})]\end{pmatrix}}\\-m^{2}(c_{1}+2c_{2}+c_{3})^{2}(kk_{2}+(k+k_{2})k_{n})\end{Bmatrix}}>0$

(21)

First we assume all dampers are positive and springs $\displaystyle k_{1},k_{2}$ have positive stiffnesses while spring $\displaystyle k_{n}$ can have negative stiffness.

$\displaystyle c_{1},c_{2},c_{3}>0\quad {\text{and}}\quad k_{1},k_{2}>0$

(22)

With these conditions, the first inequality of (20) is satisfied. Now we will prove that if the third of (20) is also satisfied:

$\displaystyle k_{n}>{\frac {-kk_{2}}{k+k_{2}}}$

(23)

then the second condtion of (20) and condition (21) automatically satisfied.

Indeed, (23) infers:

$\displaystyle {\begin{aligned}&k_{n}+k>{\frac {-kk_{2}}{k+k_{2}}}+k={\frac {k^{2}}{k+k_{2}}}>0\\&k_{n}+k_{2}>{\frac {-kk_{2}}{k+k_{2}}}+k_{2}={\frac {k_{2}^{2}}{k+k_{2}}}>0\\&a_{3}=c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})>0\\\end{aligned}}$

(24)

Now, determine

$\displaystyle {\begin{aligned}a_{1}a_{2}-a_{0}a_{3}&=m(c_{1}+2c_{2}+c_{3})[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}+(k+k_{2}+2k_{n})m]-m^{2}[c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})]\\&=m(c_{1}+2c_{2}+c_{3})[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}]+m^{2}[(c_{1}+2c_{2}+c_{3})(k+k_{2}+2k_{n})-c_{1}(k_{2}+k_{n})-c_{2}(k+k_{2})-c_{3}(k+k_{n})]\\&=m(c_{1}+2c_{2}+c_{3})[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}]+m^{2}[(c_{1}(k+k_{n})+c_{2}(k+k_{2}+4k_{n})+c_{3}(k_{2}+k_{n}))\\\end{aligned}}$

(25)

to see that:

$\displaystyle {\begin{aligned}a_{3}(a_{1}a_{2}-a_{0}a_{3})-a_{1}^{2}a_{4}&=a_{3}m(c_{1}+2c_{2}+c_{3})[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}]+m^{2}\underbrace {\begin{Bmatrix}a_{3}[c_{1}(k+k_{n})+c_{2}(k+k_{2}+4k_{n})+c_{3}(k_{2}+k_{n})]\\-(c_{1}+2c_{2}+c_{3})^{2}[kk_{2}+(k+k_{2})k_{n}]\end{Bmatrix}} _{\mathbf {P} }\end{aligned}}$

(26)

The first term (26) is obviously positive, the second term is:

$\displaystyle \mathbf {P} ={\begin{Bmatrix}{\begin{bmatrix}c_{1}(k_{2}+k_{n})+c_{2}(k+k_{2})+c_{3}(k+k_{n})\end{bmatrix}}{\begin{bmatrix}c_{1}(k+k_{n})+c_{2}(k+k_{2}+4k_{n})+c_{3}(k_{2}+k_{n})\end{bmatrix}}\\-(c_{1}+2c_{2}+c_{3})^{2}{\begin{bmatrix}kk_{2}+(k+k_{2})k_{n}\end{bmatrix}}\end{Bmatrix}}$

(27)

After some operations to rewrite (27) in the following form, which is also positive:

$\displaystyle \mathbf {P} =c_{1}^{2}k_{n}^{2}+c_{2}^{2}(k-k_{2})^{2}+c_{3}^{2}k_{n}^{2}+c_{1}c_{2}{\begin{bmatrix}(k_{2}+k_{n}-k)^{2}+3k_{n}^{2}\end{bmatrix}}+c_{2}c_{3}{\begin{bmatrix}(k+k_{n}-k_{2})^{2}+3k_{n}^{2}\end{bmatrix}}+c_{3}c_{1}{\begin{bmatrix}(k-k_{2})^{2}+2k_{n}^{2}\end{bmatrix}}>0$

(28)

Finally we conlcude that for the discrete system above stable, the conditions are (22) and (23). These conditions also reveal that adding positive dampers does not change the stability requirements of the system.